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Math-2 Review Unit 3 (Quadratic Equations)
Math-2
Review Unit 3
(Quadratic Equations)
Analyzing the graph1. Where is the function increasing?
2. Where is the function decreasing?
3. Is the function even?
4. Are there any “extrema”? If so, what type are they?
5. How does it relate to its “parent function”?
6. What is the “end behavior” of the graph?? , as yx? , as yx
7. What is the “domain” of the graph?
8. What is the “range” of the graph?
9. What is the “average rate of change” between two given values of ‘x’?
What more are we trying to learn?
We will learn some tools
that will help us to find
where graphs cross the
x-axis.
What is the y-value of an x-intercept?
(x, f(x)=0)
(-5, 0) (-2, 0) (2, 0) (5, 0)
Your turn: Factor
36 2024 xx )56(4 33 xx
)3)(2( xxMultiply the two binomials
What method did you use?
Arrows
FOIL
Distributive Property
(twice)
)5)(4( xx
)5*4()54(2 xx
2092 xx
Try the following:
1662 xx )2)(8( xx
1892 xx )3)(6( xx
18246 2 xx )34(6 2 xx
)3)(1(6 xx
“the difference of two squares”12 x
Two numbers multiplied = (-1)
and added = 0102 xx
)1)(1(
)1)(1( xx
Which of the following are NOT conjugate pairs?
)4)(4( xx
)5)(5( xx
)23)(23( xx
)5)(5( xx
Yes, they are.
Yes, they are.
NO, they are NOT.
Yes, they are.
)1)(1( xx
Your turn: Multiply the conjugate pairs.
42 x )2)(2( xx
Can we use this as a pattern in order to
factor the difference of two squares?
32 x
)2)(2( xx
)3)(3( xx
)4)(4( xx
ax
A “nice” one.
12 x
22 x
50 2 x 5x
ax 20
)5)(5(0 xx
))((0 axax
Factor23 24 xxy
)1)(2( 22 xxy
20 2 x 10 2 x
2x 11 x
))(( ixix “i” terms “cancel”
Multiply this out:2x xi xi 2i
i squared = -12x )1(
2x 1
12 xy Does the graph cross the x-axis?
How many “zeroes” does it have?
Are the “zeroes” real #’s or imaginary #’s?
Can you explain why the “sum of squares”
has imaginary zeroes?
12 xy
?42 x )2)(2( ixix
Factor the following:
))((12 ixixx
22x
?32 x
)2)(2( ixix
)3)(3( ixix
)4)(4( ixix
))((2 aixaixax
Can you see the pattern?
?72 x )7)(7( ixix
General form.
15132 2 xx15132 2 xx
)5)(32( xx
)5(3)5(2 xxx
Group the first two and last two terms
153102 2 xxx
This tells us to break
13x into 10x + 3x
3015*2
13310
3*1030 )153()102( 2 xxx
Factor out the common factors
Factor out the common factors
What are the factors of
30 that add up to 13?
)5)(32( xxyWhat are the zeroes?
10134 2 xx
)2)(54( xx
)2(5)2(4 xxx
Group the first two and last two terms
10584 2 xxx
This tells us to break
13x into 8x + 5x
4010*4
1358
5*840 )105()84( 2 xxx
Factor out the common factors
Factor out the common factors
10134 2 xx
What are the zeroes?
)2)(54( xxy
Different forms of the square function.
Vertex form (“transformation form”)
cbxaxxf 2)(
khxaxf 2)()(
Standard (polynomial) form
Intercept Form (x-intercept)
))(()( qxpxaxf
5)4(3)( 2 xxf
1572)( 2 xxxf
)4)(3(2)( xxxf
1st we need to figure out what number to add to “Complete the Square”
2x
x4 4
4
2xx
x
x8
8
x44
x
x
16
2
2
8
x
x42x
2x x8
Converting a standard form into a vertex form.
1262 xxy
khxay 2)(
What number is needed
to complete the square?
2
2
6
9
An alternative way is
add 9 and subtract 9.129962 xxy
)129(962 xxyNotice the perfect
square trinomial!!!
Convert the perfect square trinomial
to the square of a binomial3)3( 2 xy
Vertex form!!!!!!
9)1(0 2 x
x 31
2)1(9 x
khxay 2)(
Vertex form extract a square root.
9)1( 2 xy
Isolate the squared term
Let y = 0
13 x
2,4 x
“Extract a square root”
Solve for ‘x’ simplify
2)1(9 x
“Isolate the square, undo the square”
Find ‘zeroes’ by completing the square
(1) Convert to vertex form: 362 xxy
39962 xxy
(2) Set y = 0
(3) Isolate the square, undo the square
2)3(12 x
312 x
123x
123x
2
2
6
9
12)3( 2 xy
12)3(0 2 x2)3(12 x
123x
20
Your turn: Find the “zeroes” by “Extracting a square root”
21 xy
522 xy
12432 xy
18722 xy
cbxaxy 2
Quadratic Formula: gives the zeroes (x-intercepts if they are
real numbers) to ANY quadratic equation in standard form.
a
acb
a
bx
2
4
2
2
I like this version the best!
a
acbbx
2
42
But it is usually written as:
2
32
2
4x
2
324x
Simplify this:Simplify this:
a = 1
Solve using the Quadratic formula.
152 xxy
cbxaxy 2
b = 5 c = 1
)1(2
)1)(1(4)5(
)1(2
)5(2
x
If the quadratic CANNOT be factored, the solutions are “ugly.”
) (2
) )( (4) (
) 2(
) (2
x
2
21
2
5
x
a
acb
a
bx
2
4
2
2
Area of a Rectangle
9”
P = 2L + 2W
9”9”9”9”9”9”9”9”9”9”9”
length
width
Perimeter: the distance around the rectangle.
Area = L * W
You have 100 feet of fence.
100 = 2L + 2W
50 = L + Wx
???
50 – L = W
50 – x = W
50 – x Area = L * W
Area = x(50 – x)
Area of a Rectangle x
50 – x
Area = L * W
100 = x(50 – x)
A = x(50 – x)
What is the length that gives
the maximum area?
x = 25
x = 25
What is the maximum area?
A = x(50 – x)
A = 25(50 – 25)
A = (25)² = 625
Projectile Motion Problem
00
216)( stvtth
An object is launched vertically upward from the top of a 20
foot building at an initial velocity of 310 ft. per second.
a. When will the object be at its maximum height?
b. When will the object fall to the ground?
2031016)( 2 ttth
a) Find the vertex.
b) Find the x-intercept.
Projectile Motion Problem
00
216)( stvtth
An object is launched vertically upward from the ground at
an initial velocity of 450 ft per second.
b. When will the object reach 2500 feet?
ttth 45016)( 2
Find the (time, height) pairs
points of intersection.
2500)( th
),( ht 2500) ,( 1t 2500) ,(, 2t1t 2t
317616)( 2 ttthGroup A
Group B. Group C
Time(sec) 0 2 4 6 8 10
Height (ft) 3 256 381 377 246 0
What other methods could they
have used to give their data?
(x, y) pairs mapping
Which method is the best in helping you understand the
relationship between the time and height of the rockets?
graph
Which method would be the best in finding the height at a specific
time (for example: h = ? when t = 5 seconds? Equation.
The following table shows the height of an arrow at various times. Find the appropriate regression model.
T (sec)
0 1 2 3 4 5 6 7 8
H(meters)
2 77 143 199 245 282 309 327 335
When will the arrow reach its maximum height?
When will the arrow hit the ground?
2808.4)( 2 ttth
sec3.8t
sec7.16t
